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## Abstract

Investigated here is the space–time estimation or statistical interpolation of a global variable based on a few observations. Estimation of a global data field is essentially a problem of optimally estimating spherical harmonic expansion coefficients. The optimal estimation technique used here is similar to that in Kim et al. An important exception is that cyclostationary empirical orthogonal functions (CSEOFs) are used to develop the estimation technique instead of regular empirical orthogonal functions (EOFs). The use of CSEOFs is motivated by the fact that many climatic variables are (approximately) cyclostationary. That is, the statistics of a climatic variable vary periodically with a distinct nested periodicity. The developed technique is applied to estimating the global field of monthly surface temperature anomalies, which is a notable example of cyclostationary processes. The CSEOFs, an essential ingredient for formulating a cyclostationary estimation technique, account for the monthly variation of the surface temperature statistics, namely much larger variance in the winter than in the summer. Further, cyclostationary statistics contain information on how different months are correlated. This allows one to use all 12 months of measurements, thereby optimizing the estimation technique both in space and time. As the test results indicate, estimation error is much reduced when using the cyclostationary technique.

## Abstract

Investigated here is the space–time estimation or statistical interpolation of a global variable based on a few observations. Estimation of a global data field is essentially a problem of optimally estimating spherical harmonic expansion coefficients. The optimal estimation technique used here is similar to that in Kim et al. An important exception is that cyclostationary empirical orthogonal functions (CSEOFs) are used to develop the estimation technique instead of regular empirical orthogonal functions (EOFs). The use of CSEOFs is motivated by the fact that many climatic variables are (approximately) cyclostationary. That is, the statistics of a climatic variable vary periodically with a distinct nested periodicity. The developed technique is applied to estimating the global field of monthly surface temperature anomalies, which is a notable example of cyclostationary processes. The CSEOFs, an essential ingredient for formulating a cyclostationary estimation technique, account for the monthly variation of the surface temperature statistics, namely much larger variance in the winter than in the summer. Further, cyclostationary statistics contain information on how different months are correlated. This allows one to use all 12 months of measurements, thereby optimizing the estimation technique both in space and time. As the test results indicate, estimation error is much reduced when using the cyclostationary technique.

## Abstract

Considered in this study is a cyclostationary generalization of an EOF-based prediction method. While linear statistical prediction methods are typically optimal in the sense that prediction error variance is minimal within the assumption of stationarity, there is some room for improved performance since many physical processes are not stationary. For instance, El Niño is known to be strongly phase locked with the seasonal cycle, which suggests nonstationarity of the El Niño statistics. Many geophysical and climatological processes may be termed cyclostationary since their statistics show strong cyclicity instead of stationarity. Therefore, developed in this study is a cyclostationary prediction method. Test results demonstrate that performance of prediction methods can be improved significantly by accounting for the cyclostationarity of underlying processes. The improvement comes from an accurate rendition of covariance structure both in space and time.

## Abstract

Considered in this study is a cyclostationary generalization of an EOF-based prediction method. While linear statistical prediction methods are typically optimal in the sense that prediction error variance is minimal within the assumption of stationarity, there is some room for improved performance since many physical processes are not stationary. For instance, El Niño is known to be strongly phase locked with the seasonal cycle, which suggests nonstationarity of the El Niño statistics. Many geophysical and climatological processes may be termed cyclostationary since their statistics show strong cyclicity instead of stationarity. Therefore, developed in this study is a cyclostationary prediction method. Test results demonstrate that performance of prediction methods can be improved significantly by accounting for the cyclostationarity of underlying processes. The improvement comes from an accurate rendition of covariance structure both in space and time.

## Abstract

Extraction of the accurate annual cycle in the tropical Pacific sea surface temperature field is addressed as a demonstration of the utility of a new technique called the cyclostationary empirical orthogonal function (CSEOF) analysis. The strength of the annual cycle has fluctuated roughly by 15% in the past 30 yr (1970–99), and this fluctuation includes swings every 4–6 yr. Accurate extraction of the detailed structure and temporal modulation of the annual cycle was owing to the application of CSEOF analysis, the concept of which has been introduced and compared to the conventional EOF analysis.

## Abstract

Extraction of the accurate annual cycle in the tropical Pacific sea surface temperature field is addressed as a demonstration of the utility of a new technique called the cyclostationary empirical orthogonal function (CSEOF) analysis. The strength of the annual cycle has fluctuated roughly by 15% in the past 30 yr (1970–99), and this fluctuation includes swings every 4–6 yr. Accurate extraction of the detailed structure and temporal modulation of the annual cycle was owing to the application of CSEOF analysis, the concept of which has been introduced and compared to the conventional EOF analysis.

## Abstract

Many climatic and geophysical processes are cyclostationary and exhibit appreciable cyclic (monthly, daily, etc.) variation of their statistics in addition to interannual fluctuations. Utilization of this nested variation of statistics will lead to a better chance of detecting a signal in such a varying background noise field, especially when the signal is strongly phase locked with the nested cycle. In this study, a detection technique is constructed in terms of cyclostationary empirical orthogonal functions, which take the nested periodicity of noise statistics into account. To investigate the improved performance of the cyclostationary approach the developed algorithm is applied to three specific detection examples: El Niño, greenhouse warming, and sunspot fluctuations. In all the test cases, signal-to-noise ratio is raised between 2% and 43% compared with that of a stationary detection technique. The variation of signal strength when a detection filter is constructed based on a different section of modeled noise is within the range of mean signal-to-noise ratio for small to moderate signals. There is a significant variation, however, of signal strength when a detection filter is constructed based on a different model dataset. This implies that model discrepancy is a more important factor than sampling error for the accuracy of the detection method and that climate models need to be improved further in their noise statistics.

## Abstract

Many climatic and geophysical processes are cyclostationary and exhibit appreciable cyclic (monthly, daily, etc.) variation of their statistics in addition to interannual fluctuations. Utilization of this nested variation of statistics will lead to a better chance of detecting a signal in such a varying background noise field, especially when the signal is strongly phase locked with the nested cycle. In this study, a detection technique is constructed in terms of cyclostationary empirical orthogonal functions, which take the nested periodicity of noise statistics into account. To investigate the improved performance of the cyclostationary approach the developed algorithm is applied to three specific detection examples: El Niño, greenhouse warming, and sunspot fluctuations. In all the test cases, signal-to-noise ratio is raised between 2% and 43% compared with that of a stationary detection technique. The variation of signal strength when a detection filter is constructed based on a different section of modeled noise is within the range of mean signal-to-noise ratio for small to moderate signals. There is a significant variation, however, of signal strength when a detection filter is constructed based on a different model dataset. This implies that model discrepancy is a more important factor than sampling error for the accuracy of the detection method and that climate models need to be improved further in their noise statistics.

## Abstract

Identification of independent physical/dynamical modes and corresponding principal component time series is an important aspect of climate studies for they serve as a tool for detecting and predicting climate changes. While there are a number of different eigen techniques their performance for identifying independent modes varies. Considered here are comparison tests of eight eigen techniques in identifying independent patterns from a dataset. A particular emphasis is given to cyclostationary processes such as deforming and moving patterns with cyclic statistics. Such processes are fairly common in climatology and geophysics. Two eigen techniques that are based on the cyclostationarity assumption—cyclostationary empirical orthogonal functions (EOFs) and periodically extended EOFs—perform better in identifying moving and deforming patterns than techniques based on the stationarity assumption. Application to a tropical Pacific surface temperature field indicates that the first dominant pattern and the corresponding principal component (PC) time series are consistent among different techniques. The second mode and the PC time series, however, are not very consistent from one another with hints of significant modal mixing and splitting in some of derived patterns. There also is a detailed difference of intraannual scale between PC time series of a stationary technique and those of a cyclostationary one. This may bear an important implication on the predictability of El Niño. Clearly there is a choice of eigen technique for improved predictability.

## Abstract

Identification of independent physical/dynamical modes and corresponding principal component time series is an important aspect of climate studies for they serve as a tool for detecting and predicting climate changes. While there are a number of different eigen techniques their performance for identifying independent modes varies. Considered here are comparison tests of eight eigen techniques in identifying independent patterns from a dataset. A particular emphasis is given to cyclostationary processes such as deforming and moving patterns with cyclic statistics. Such processes are fairly common in climatology and geophysics. Two eigen techniques that are based on the cyclostationarity assumption—cyclostationary empirical orthogonal functions (EOFs) and periodically extended EOFs—perform better in identifying moving and deforming patterns than techniques based on the stationarity assumption. Application to a tropical Pacific surface temperature field indicates that the first dominant pattern and the corresponding principal component (PC) time series are consistent among different techniques. The second mode and the PC time series, however, are not very consistent from one another with hints of significant modal mixing and splitting in some of derived patterns. There also is a detailed difference of intraannual scale between PC time series of a stationary technique and those of a cyclostationary one. This may bear an important implication on the predictability of El Niño. Clearly there is a choice of eigen technique for improved predictability.

The perception of the hypothesized greenhouse effect will differ dramatically depending upon the location on the earth at which the effect is analyzed. This is due mainly to two causes: 1) the warming signal depends upon the position on the earth, and 2) the natural variability of the warming has a strong position dependence. To demonstrate these phenomena, simulations were conducted of the surface temperature field with a simple stochastic climate model that has enough geographical resolution to see the geographic dependence. The model was tuned to reproduce the geographical distribution of the present climate, including its natural variability in both the variance and the space–time correlation structure. While such effects have been discussed elsewhere with even more realistic climate models, it is instructive to actually see simulations of time series laid side by side in order to easily compare their differences and similarities. Because of the model's simplicity, the causes of the variations are easy to analyze. Not surprisingly, some realizations of the temperature for some local areas show countertrends for a period of several decades in the presence of the greenhouse warming.

The perception of the hypothesized greenhouse effect will differ dramatically depending upon the location on the earth at which the effect is analyzed. This is due mainly to two causes: 1) the warming signal depends upon the position on the earth, and 2) the natural variability of the warming has a strong position dependence. To demonstrate these phenomena, simulations were conducted of the surface temperature field with a simple stochastic climate model that has enough geographical resolution to see the geographic dependence. The model was tuned to reproduce the geographical distribution of the present climate, including its natural variability in both the variance and the space–time correlation structure. While such effects have been discussed elsewhere with even more realistic climate models, it is instructive to actually see simulations of time series laid side by side in order to easily compare their differences and similarities. Because of the model's simplicity, the causes of the variations are easy to analyze. Not surprisingly, some realizations of the temperature for some local areas show countertrends for a period of several decades in the presence of the greenhouse warming.

## Abstract

Considered here are examples of statistical prediction based on the algorithm developed by Kim and North. The predictor is constructed in terms of space–time EOFs of data and prediction domains. These EOFs are essentially a different representation of the covariance matrix, which is derived from past observational data. The two sets of EOFs contain information on how to extend the data domain into prediction domain (i.e., statistical prediction) with minimum error variance. The performance of the predictor is similar to that of an optimal autoregressive model since both methods are based on the minimization of prediction error variance. Four different prediction techniques—canonical correlation analysis (CCA), maximum covariance analysis (MCA), principal component regression (PCR), and principal oscillation pattern (POP)—have been compared with the present method. A comparison shows that oscillation patterns in a dataset can faithfully be extended in terms of temporal EOFs, resulting in a slightly better performance of the present method than that of the predictors based on the maximum pattern correlations (CCA, MCA, and PCR) or the POP predictor. One-dimensional applications demonstrate the usefulness of the predictor. The NINO3 and the NINO3.4 sea surface temperature time series (3-month moving average) were forecasted reasonably up to the lead time of about 6 months. The prediction skill seems to be comparable to other more elaborate statistical methods. Two-dimensional prediction examples also demonstrate the utility of the new algorithm. The spatial patterns of SST anomaly field (3-month moving average) were forecasted reasonably up to about 6 months ahead. All these examples illustrate that the prediction algorithm is useful and computationally efficient for routine prediction practices.

## Abstract

Considered here are examples of statistical prediction based on the algorithm developed by Kim and North. The predictor is constructed in terms of space–time EOFs of data and prediction domains. These EOFs are essentially a different representation of the covariance matrix, which is derived from past observational data. The two sets of EOFs contain information on how to extend the data domain into prediction domain (i.e., statistical prediction) with minimum error variance. The performance of the predictor is similar to that of an optimal autoregressive model since both methods are based on the minimization of prediction error variance. Four different prediction techniques—canonical correlation analysis (CCA), maximum covariance analysis (MCA), principal component regression (PCR), and principal oscillation pattern (POP)—have been compared with the present method. A comparison shows that oscillation patterns in a dataset can faithfully be extended in terms of temporal EOFs, resulting in a slightly better performance of the present method than that of the predictors based on the maximum pattern correlations (CCA, MCA, and PCR) or the POP predictor. One-dimensional applications demonstrate the usefulness of the predictor. The NINO3 and the NINO3.4 sea surface temperature time series (3-month moving average) were forecasted reasonably up to the lead time of about 6 months. The prediction skill seems to be comparable to other more elaborate statistical methods. Two-dimensional prediction examples also demonstrate the utility of the new algorithm. The spatial patterns of SST anomaly field (3-month moving average) were forecasted reasonably up to about 6 months ahead. All these examples illustrate that the prediction algorithm is useful and computationally efficient for routine prediction practices.

## Abstract

This study makes use of a simple stochastic energy balance climate model that resolves the land–sea distribution and that includes a crude upwelling-diffusion deep ocean to study the natural variability of the surface temperature in different frequency bands. This is done by computing the eigenfunctions of the space-time lagged covariance function. The resulting frequency-dependent theoretical orthogonal functions (fdTOFs) are compared with the corresponding frequency-dependent empirical orthogonal functions (fdEOFs) derived from 40 years of data. The computed and modeled eigenvalues are consistent with the difference mainly explained by sampling error due to the short observational record. The magnitude of expected sampling errors is demonstrated by a series of Monte Carlo simulations with the model. The sampling error for the eigenvalues features a strong bias that appears in the simulations and apparently in the data. Component-by-component pattern correlations between the fdEOFs and the fdTOFs vary from 0.81 to 0.28 for the first ten components. Monte Carlo simulations show that the sampling error could be an important source of error especially in the low (interannual) frequency band. However, sampling error alone cannot satisfactorily explain the difference between the model and observations. Rather, model inaccuracy and/or spatial bias of observations seem to be important sources of error. The fdTOFs are expected to be useful in estimation/prediction/detection studies.

## Abstract

This study makes use of a simple stochastic energy balance climate model that resolves the land–sea distribution and that includes a crude upwelling-diffusion deep ocean to study the natural variability of the surface temperature in different frequency bands. This is done by computing the eigenfunctions of the space-time lagged covariance function. The resulting frequency-dependent theoretical orthogonal functions (fdTOFs) are compared with the corresponding frequency-dependent empirical orthogonal functions (fdEOFs) derived from 40 years of data. The computed and modeled eigenvalues are consistent with the difference mainly explained by sampling error due to the short observational record. The magnitude of expected sampling errors is demonstrated by a series of Monte Carlo simulations with the model. The sampling error for the eigenvalues features a strong bias that appears in the simulations and apparently in the data. Component-by-component pattern correlations between the fdEOFs and the fdTOFs vary from 0.81 to 0.28 for the first ten components. Monte Carlo simulations show that the sampling error could be an important source of error especially in the low (interannual) frequency band. However, sampling error alone cannot satisfactorily explain the difference between the model and observations. Rather, model inaccuracy and/or spatial bias of observations seem to be important sources of error. The fdTOFs are expected to be useful in estimation/prediction/detection studies.

## Abstract

Air temperature anomalies, averaged over the troposphere to 200 mb and around the earth from 10°S to 10°N, lag the similarly averaged El Niño–Southern Oscillation (ENSO) atmospheric latent heating anomalies by about one month. Most of the latent heating is balanced by vertical adiabatic cooling although the zonally averaged imbalance is larger than is typical locally in the Tropics. The excess latent heating heats the atmosphere and generates a temperature anomaly. As the temperature anomaly rises, the atmosphere loses heat until the residual heating is balanced by anomalous cooling. By then the temperature anomaly is typically about 0.4°C. Analysis of the thermodynamic energy equation shows that the ENSO heat loss is highly linearly correlated with the air temperature anomaly averaged over the equatorial troposphere; that is, the adjustment to the residual anomalous heating (or cooling) is Newtonian. Consistent with the observed one-month lag, the Newtonian *e*-folding time is about 35 days. Similar results apply for latitude bands 5°S–5°N and 15°S–15°N (Newtonian cooling times of 29 and 46 days, respectively). The heat loss is mainly through meridional sensible heat flux rather than radiation. Much of the anomalous cooling is due to the mean meridional flow that diverges more temperature anomaly aloft than it converges near the surface.

## Abstract

Air temperature anomalies, averaged over the troposphere to 200 mb and around the earth from 10°S to 10°N, lag the similarly averaged El Niño–Southern Oscillation (ENSO) atmospheric latent heating anomalies by about one month. Most of the latent heating is balanced by vertical adiabatic cooling although the zonally averaged imbalance is larger than is typical locally in the Tropics. The excess latent heating heats the atmosphere and generates a temperature anomaly. As the temperature anomaly rises, the atmosphere loses heat until the residual heating is balanced by anomalous cooling. By then the temperature anomaly is typically about 0.4°C. Analysis of the thermodynamic energy equation shows that the ENSO heat loss is highly linearly correlated with the air temperature anomaly averaged over the equatorial troposphere; that is, the adjustment to the residual anomalous heating (or cooling) is Newtonian. Consistent with the observed one-month lag, the Newtonian *e*-folding time is about 35 days. Similar results apply for latitude bands 5°S–5°N and 15°S–15°N (Newtonian cooling times of 29 and 46 days, respectively). The heat loss is mainly through meridional sensible heat flux rather than radiation. Much of the anomalous cooling is due to the mean meridional flow that diverges more temperature anomaly aloft than it converges near the surface.

## Abstract

Observations show that regions of anomalous deep convective El Niño–Southern Oscillation (ENSO) heating tend to be balanced by anomalous ENSO cooling elsewhere so that, averaged around the globe from (say) 10°S to 10°N, the net anomalous heating is nearly zero. The zonally symmetric heating is weak because it is approximately proportional to vertical velocity that, when averaged over a constant pressure surface *S* around the earth from 10°S to 10°N, is nearly zero. The horizontally averaged vertical velocity over *S* is small because the net horizontal geostrophic convergent flow across 10°S and 10°N is zero.

Although the zonally symmetric ENSO heating is weak, the observed ENSO tropospheric air temperature anomaly has a large zonally symmetric component. Past work has shown that with weak momentum and thermal damping, Kelvin and Rossby waves can travel around the earth without significant loss of amplitude so that a zonally symmetric response is favored. This physical interpretation depends on knowing temperature and momentum anomaly damping times over the depth of the troposphere. Such times are not well known. Here a Gill tropical atmospheric model is generalized to include realistic surface friction and so theoretically estimate a frictional spindown time. Using this spindown time (approximately 3 weeks), together with an estimate of the Newtonian cooling time (1 month) the authors show, in agreement with observations, that the extremely weak zonally symmetric heating anomaly generates a symmetric air temperature anomaly comparable to the asymmetric one.

## Abstract

Observations show that regions of anomalous deep convective El Niño–Southern Oscillation (ENSO) heating tend to be balanced by anomalous ENSO cooling elsewhere so that, averaged around the globe from (say) 10°S to 10°N, the net anomalous heating is nearly zero. The zonally symmetric heating is weak because it is approximately proportional to vertical velocity that, when averaged over a constant pressure surface *S* around the earth from 10°S to 10°N, is nearly zero. The horizontally averaged vertical velocity over *S* is small because the net horizontal geostrophic convergent flow across 10°S and 10°N is zero.

Although the zonally symmetric ENSO heating is weak, the observed ENSO tropospheric air temperature anomaly has a large zonally symmetric component. Past work has shown that with weak momentum and thermal damping, Kelvin and Rossby waves can travel around the earth without significant loss of amplitude so that a zonally symmetric response is favored. This physical interpretation depends on knowing temperature and momentum anomaly damping times over the depth of the troposphere. Such times are not well known. Here a Gill tropical atmospheric model is generalized to include realistic surface friction and so theoretically estimate a frictional spindown time. Using this spindown time (approximately 3 weeks), together with an estimate of the Newtonian cooling time (1 month) the authors show, in agreement with observations, that the extremely weak zonally symmetric heating anomaly generates a symmetric air temperature anomaly comparable to the asymmetric one.